DATA ASSIMILATION AT LOCAL SCALE TOIMPROVE CFD SIMULATIONS OF DISPERSION

AROUND INDUSTRIAL SITES AND URBANNEIGHBOURHOODS

C. Defforge1, M. Bocquet1, R. Bresson1, P. Armand2,and B. Carissimo1

1CEREA, Joint laboratory Ecole des Ponts ParisTech / EDF R&D, Universite Paris-Est, Marne-la-Vallee, France

2CEA, DAM, DIF, F-91297 Arpajon, France

Harmo18 - Mathematical problems in air quality modelling

IntroductionContextIntroduction to data assimilation

MethodsShallow water modelBack and forth nudgingIterative ensemble Kalman smoother

ResultsExperimentsBFN resultsIEnKS results

Conclusions & Perspectives

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 1 / 14

MICRO-METEOROLOGY APPLICATIONSI Dispersion in built up environment

City of Toulouse MUST experiment

I Estimation of local wind fields

↔

Turbine wakes

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 2 / 14

CONTEXT

I Atmospheric dispersion modelling requires meteorological inputs (wind,turbulence, etc.)

I Local wind fields (urban neighbourhoods, surroundings of industrial sites,etc.) have very complex structures ⇒ difficult to simulate with CFD

I CFD simulations could be improved using available observations

I Objective: Develop local-scale data assimilation methods

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 3 / 14

LOCAL CFD SIMULATIONSMesoscale simulations

(e.g. WRF, ALADIN)∆x ≈ 10km, ∆z ≈ 10m

L ≈ 3000km, LU ≈ 7 days

x

t

IC

Boundaryconditions

Local simulations(e.g. Code Saturne)

∆x ≈ 10m, ∆z ≈ 1m

L ≈ 5km, LU ≈ 17min

x

t

BC

Observations

Data assimilation

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 4 / 14

INTRODUCTION TO DATA ASSIMILATION

I za: analysis = best estimate of control variables z, given all availableinformation

I model M,I observations yo ,I prior knowledge zb,I etc.

I Nudging: add relaxation term to dynamical equationsI Back and forth nudging (BFN)

I Filtering methods (e.g. Kalman filter) and Variational methods (e.g.3D-Var)

I Ensemble variational methods: iterative ensemble Kalmansmoother/filter (IEnKS, IEnKF)

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 5 / 14

SHALLOW WATER MODEL

I’Level’ models ⇐⇒ ’Layer’ models

Vertical finite-difference approximation Multi-layer SWE

I Vertically averaged equations: ∂X∂t + M∂X

∂x = S

X =

(hu

), M =

(u hg ′ u

), S =

(0

−g ′ ∂zf∂x

), and g ′ : reduced gravity

0 1000 2000 3000 4000 5000 6000 7000 8000x [m]

0

250

500

750

1000

1250

1500

h[m

]

zf

hu

free atmosphere

u [m/s]4.0

5.6

7.2

8.8

10.4

Simulation with 1D shallow water model over topography.

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 6 / 14

BACK AND FORTH NUDGING ALGORITHM

Iterative algorithm of forward and backward integrations with nudging 1:

(F)∂Xf

k∂t + Mf ∂Xf

k∂x = S + K

[yo −H(Xf

k)]

for 0 ≤ t ≤ T , δt > 0

(B)∂Xb

k∂t + Mb ∂Xb

k∂x = S− K

[yo −H(Xb

k)]

for T ≥ t ≥ 0, δt < 0

forward (f) or backward (b)

k: BFN iteration

Observation operator

1Auroux and Blum (2005, 2008); Auroux et al. (2013)C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 7 / 14

BOUNDARY CONDITIONS FOR BFN ALGORITHM

0 2000 4000 6000 8000x [m]

0

500

1000

1500

h[m

]

uL hR

u [m/s]4.0

5.6

7.2

8.8

10.4

Forward

0 2000 4000 6000 8000x [m]

0

500

1000

1500

h[m

]

xo0 xo1 xo2 xo3 xo4

u [m/s]4.0

5.6

7.2

8.8

10.4

0 2000 4000 6000 8000x [m]

0

500

1000

1500

h[m

]

˜uRhL

u [m/s]−12.0

−10.4

−8.8

−7.2

−5.6

u = −uukR = −uk(x = L)

hkL = hk(x = 0)

0 2000 4000 6000 8000x [m]

0

500

1000

1500

h[m

]

xo0 xo1 xo2 xo3 xo4

u [m/s]−12.0

−10.4

−8.8

−7.2

−5.6

Backward

u = −uuk+1L = −uk(x = 0)

hk+1R = hk(x = L)

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 8 / 14

ITERATIVE ENSEMBLE KALMAN SMOOTHER 1

I Cost function:J = ‖distance to prior‖P−1 + ‖distance to observations‖R−1

I Ensemble method → estimation of error covariance matrices

I Iterative minimisation of J with Gauss-Newton algorithm

I 2 cycles of IEnKS algorithm:

Analysisat tk−1

Lyk−L yk−1

zak−L

Analysisat tk

yk

zbk−L+1

=M(zak−L)

zak−L+1

1Sakov et al. (2012); Bocquet and Sakov (2014)C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 9 / 14

EXPERIMENTS

True BCsutL = 5.5m/s

htR = 617m

5 observations

0 2000 4000 6000 8000x [m]

0

500

1000

1500

h[m

]

uo0 uo1 uo2 uo3 uo4

4.0

5.6

7.2

8.8

10.4

u[m

/s]

Referencesimulation

A priori BCs

ubL = 4.4m/s

hbR = 617m

Experiment 1

Experiment 2

perfect obs.

noisy obs.

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 10 / 14

BFN RESULTSI K = K = kHT where k∆t = 0.1I Convergence in ∼ 5 iterations

0 50 100 150 200 250 300x [m]

4

5

6

7

8

9

10

11

12

u[m

/s]

BackgroundTrue stateObservationsBFN: 1st iterationBFN: 5th iterationBFN: 10th iteration

Exp. 1: Perfect observations

0 50 100 150 200 250 300x [m]

4

5

6

7

8

9

10

11

12

u[m

/s]

Exp. 2: Noisy observations

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 11 / 14

IEnKS RESULTSI Background ensemble: 3 membersI P = I and R = 0.1II Fast convergence (2-3 iterations)

0 50 100 150 200 250 300x [m]

3

4

5

6

7

8

9

10

u[m

/s]

BackgroundTrue stateObservationsMembersIEnKS

Exp. 1: Perfect observations

0 50 100 150 200 250 300x [m]

3

4

5

6

7

8

9

10

u[m

/s]

Exp. 2: Noisy observations

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 12 / 14

CONCLUSIONS & PERSPECTIVES

I Both BFN algorithm and IEnKS help correcting BCs

I IEnKS more efficient here (less model integrations)

I Next steps:I More complex cases:

I SW model: 2DI Code Saturne: Vertical profiles of u

I Localization or reduction of control vector size (e.g. principal componentanalysis)

I Realistic cases with Code Saturne (buildings, obstacles, etc.)

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 13 / 14

THANK YOU FOR YOUR ATTENTION

REFERENCES

Auroux, D., P. Bansart, and J. Blum, 2013: An evolution of the back andforth nudging for geophysical data assimilation: application to burgersequation and comparisons. Inverse Probl. Sci. Eng., 21, 399–419.

Auroux, D., and J. Blum, 2005: Back and forth nudging algorithm for dataassimilation problems. Comptes Rendus Math., 340, 873–878.

Auroux, D., and J. Blum, 2008: A nudging-based data assimilation method:the Back and Forth Nudging (BFN) algorithm. Nonlin. Process. Geophys.,15, 305–319.

Bocquet, M., and P. Sakov, 2014: An iterative ensemble Kalman smoother.Quart. J. Royal Meteor. Soc., 140, 1521–1535.

Sakov, P., D. S. Oliver, and L. Bertino, 2012: An Iterative EnKF for StronglyNonlinear Systems. Mon. Wea. Rev., 140, 1988–2004.

C. Defforge (CEREA) et al. Data assimilation at local scale to improve CFD simulations 14 / 14

IEnKS ALGORITHM

Background ensemble: E0 = z(0)0 1T + A0. Initialisation: w = 0

(BCs)mean anomalies

w z′0= z

(0)0 1T + A0w

E′0 E′LModel yL

Obs.operator

dy

Obs.yo

∇J , H∆w

+

wa, za0 = z(0)0 + A0wa, Ea

0 = za01T + A0H−1/2

until ‖∆w‖ < e

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